In his latest book 'The crash of 2016' author Thom Hartmann describes a story he once read in 'Amazing Stories'.

He doesn't mention a title or the author. Here is how he writes about it:

I was around ten years old, and a total science fiction junkie.
Amazing Stories—a pulp magazine of science fiction short stories,
commentary, and science news—arrived every month, and that meant that
on that day all homework and play were forgotten. These were some of
the most brilliant stories, written for a penny or so a word by people
such as Robert Silverberg, Jack Vance, Poul Anderson, and Frank
Herbert, who would go on to become the biggest names in the field of
Fantasy and Sci-Fi.

And one particular story haunted me for years. Eight years later, when
the first men landed on the Moon, that story was brought back to me as
if I’d read it the day before.

In the story, a group of astronauts are finally, for the first time,
going to blast off from Earth and circle the Moon. It had never been
done before, and because our moon doesn’t itself rotate on its axis
like we do relative to the Sun, but instead circles us with a single
side always facing us, nobody had ever before seen the “dark back side
of the Moon.” These men would be the first in human history to do so.

The rocket roared to life from the launchpad, the astronauts chatting
with Mission Control as they hurtled toward the Moon. When they got
close enough, they let the Moon’s light gravity grab their space
capsule and, with a few deft rocket thrusts, they put themselves into
orbit around it. They were unbelievably excited, as were the NASA
folks on the ground, chattering back and forth about what they hoped
and expected to see.

Was the dark side of the Moon identical to the front, or was it more
mountainous? (We now know it is the latter.) Might there be frozen
water there, since it has a different exposure to the Sun? Might there
even be wreckage there from ancient astronauts, or something else
exotic like that? Anything was possible!

As the astronauts began their turn around to the back side of the
Moon, Mission Control told them they’d lose communication because the
giant mass of the Moon would block their signals, so they prepared for
radio silence. But as they made the turn, they could still hear
Mission Control.

What they saw as they glimpsed behind the face of the Moon we see on
full-moon nights brought an audible gasp from all of them.

One picked up the microphone to radio the ground and tell the horrible
story. Another knocked the microphone from his hand with a warning
gesture. They continued to circle around the back side of the Moon,
and the view became even clearer, ever more undeniable.

“Should we report it?” A debate—virtually a fistfight—broke out in the
capsule. What would it mean for earthlings? What if nobody believed
them? What if they were quarantined upon returning and imprisoned or
sent to a mental hospital? I remember what they saw that so horrified
the astronauts.

The back side of the Moon was missing. They were looking into the
concave half ball of the front side of the Moon, which was made of
canvas stretched over an elaborate superstructure of two-by-fours,
nailed together like the scaffolds that held up roller coasters from
that era.

1 Answer
1

But Oldbury still looked at the Moon. It was terribly close and now the surface was moving quickly. They were starting the swing in earnest and Oldbury’s scream was high-pitched. “Look! Lookathat!” His pointing finger was stiff with terror.

...

“You saw it. You saw the hidden side of the Moon as we went past and you saw there wasn’t any! Good Lord, just sticks, just big beams holding up six million square miles of canvas. I swear it, canvas!”

However it was just an experiment to see how the astronauts would cope with the stress of the flight. They weren't meant to see the far side of the fake moon but the experiment went wrong:

“We learned a great deal, so the experiment was a success in its way. Now listen—the controls of the ship were designed to go wrong when they did in order to test your reaction to emergency conditions after several days of travel strain. The breakdown was timed for the simulated swing about the Moon, which we were going to switch about so that you could see it from a new angle on the return trip. You weren’t intended to see the other side and so we didn’t build the other side. Call it economy. This test alone cost fifty million dollars and it’s not easy to get appropriations.”

Nilsson added bitterly, “Except that the shut-off switch on the ’scope didn’t shut off in time. A valve caught. You saw the unfinished back of the Moon and we had to stop the ship to prevent—”

Or was it ...

“Our unmanned rockets, three of them—the information devices on each stopped transmitting just before the boomerang swing and not one returned. Sometimes I just wonder—” “Shut up" said Nilsson fiercely.

As a comment points out, the story has not been published in Amazing Stories, or at least not according to IFSDB. The first publication was in Galaxy in 1957.